This is bad advice. Unless your current US phone is from the dark ages, it will work in Europe. Buy a prepaid SIM card when you arrive, put it in your phone and text away. If you set up an account you will be able to top up the SIM any time by SMS or a call, but you can buy an anonymous SIM if you prefer.
The SIM will work in any European country, and text charges will be only minimally more expensive as you move out of the country you purchased the SIM in (and will certainly be less expensive than buying a new SIM in each country; that's just a complete waste of time, not to mention a pain in the arse - unless you enjoy visiting mobile phone shops and filling out forms?). Stick to texting and have people call you in the hotel or on other land lines (or look for internet cafes to call from) and don't accept any calls from the US on your phone if you are outside the country where you purchase your SIM - there is a charge to you for accepting those calls.
A word of warning - do NOT use texting apps, and make sure that you have data roaming switched off on your phone, because as soon as you move from the country you purchase the SIM in, you will be hit with international roaming charges for data. Data charges are not covered by the same laws as mobile phone calls and SMS - you will burn your credit in no time.
You are actually visiting a year too soon - in 2014 "international" calls within the EU will disappear, and all mobile phone calls will be charged at one rate wherever you are in the EU. You will still pay international rates to call the US, but you won't be charged extra because e.g. your phone plan was bought in France and you are in Sweden.